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Fused Angles for Body Orientation Representation Philipp Allgeuer and Sven Behnke Institute for Computer Science VI Autonomous Intelligent Systems University of Bonn

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Nov 18, 2014Fused Angles for Body Orientation Representation2 Motivation What is a rotation representation? A parameterisation of the manifold of all rotations in three-dimensional Euclidean space Why do we need them? To perform calculations relating to rotations Existing rotation representations? Rotation matrices, quaternions, Euler angles, … Why develop a new representation? Desired for the analysis and control of balancing bodies in 3D (e.g. a biped robot)

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Nov 18, 2014Fused Angles for Body Orientation Representation3 Problem Definition The problem: Find a representation that describes the state of balance in an intuitive and problem-relevant way, and yields information about the components of the rotation in the three major planes (xy, yz, xz) Orientation A rotation relative to a global fixed frame Relevant as an expression of attitude for balance Environment Fixed, z-axis points ‘up’ (i.e. opposite to gravity)

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Nov 18, 2014Fused Angles for Body Orientation Representation4 Problem Definition The solution: Fused angles (and the intermediate tilt angles representation)

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Nov 18, 2014Fused Angles for Body Orientation Representation5 Uses of Fused Angles to Date Attitude Estimator [1] [2] Internally based on the concept of fused angles for orientation resolution NimbRo ROS Soccer Package [4] [5] Intended for the NimbRo-OP humanoid robot Fused angles are used for state estimation and the walking control engine Matlab/Octave Rotations Library [6] Library for computations related to rotations in 3D (supports both fused angles and tilt angles)

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Nov 18, 2014Fused Angles for Body Orientation Representation6 Existing Representations Rotation matrices Quaternions Euler angles Axis-angle Rotation vectors Vectorial parameterisations

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Nov 18, 2014Fused Angles for Body Orientation Representation7 Containing set: Parameters:3 ⇒ Minimal Constraints:None Singularities:Gimbal lock at the limits of β Features: Splits rotation into a sequence of elemental rotations, numerically problematic near the singularities, computationally inefficient Intrinsic ZYX Euler Angles

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Nov 18, 2014Fused Angles for Body Orientation Representation8 Intrinsic ZYX Euler Angles Relevant feature: Quantifies the amount of rotation about the x, y and z axes ≈ in the three major planes Problems: Proximity of both gimbal lock singularities to normal working ranges, high local sensitivity Requirement of an order of elemental rotations, leading to asymmetrical definitions of pitch/roll Unintuitive non-axisymmetric behaviour of the yaw angle due to the reliance on axis projection

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Nov 18, 2014Fused Angles for Body Orientation Representation9 Tilt Angles Rotation G to B ψ= Fused yaw γ= Tilt axis angle α= Tilt angle

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Nov 18, 2014Fused Angles for Body Orientation Representation10 Tilt Angles Features: Geometrically and mathematically very relevant Intuitive and axisymmetric definitions Drawbacks: γ parameter is unstable near the limits of α!

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Nov 18, 2014Fused Angles for Body Orientation Representation11 Fused Angles Rotation G to B Pure tilt rotation! θ= Fused pitch φ= Fused roll h= Hemisphere

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Nov 18, 2014Fused Angles for Body Orientation Representation12 Fused Angle Level Sets

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Nov 18, 2014Fused Angles for Body Orientation Representation13 Fused Angle Level Sets

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Nov 18, 2014Fused Angles for Body Orientation Representation14 Intersection of Level Sets

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Nov 18, 2014Fused Angles for Body Orientation Representation15 Fused Angles Condition for validity: Sine sum criterion Set of all fused angles:

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Nov 18, 2014Fused Angles for Body Orientation Representation16 Sine Sum Criterion

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Nov 18, 2014Fused Angles for Body Orientation Representation17 Mathematical Definitions By analysis of the geometric definitions:

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Nov 18, 2014Fused Angles for Body Orientation Representation18 Representation Conversions Fused angles ⇔ Tilt angles Surprisingly fundamental conversions Representations intricately linked Fused angles ⇔ Rotation matrices, quaternions Simple and robust conversions available Tilt angles ⇔ Rotation matrices, quaternions Robust and direct conversions available Simpler definition of fused yaw arises Refer to the paper

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Nov 18, 2014Fused Angles for Body Orientation Representation19 Tilt axis angle γ has singularities at α = 0, π …but has increasingly little effect near α = 0 Fused yaw ψ has a singularity at α = π Unavoidable due to the minimality of (ψ,θ,φ) As ‘far away’ from the identity rotation as possible Define ψ = 0 on this null set Fused yaw and quaternions Properties

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Nov 18, 2014Fused Angles for Body Orientation Representation20 Properties Inverse of a fused angles rotation Special case of zero fused yaw

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Nov 18, 2014Fused Angles for Body Orientation Representation21 Matlab/Octave Rotations Library https://github.com/AIS-Bonn/matlab_octave_rotations_lib https://github.com/AIS-Bonn/matlab_octave_rotations_lib Thank you for your attention!

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Nov 18, 2014Fused Angles for Body Orientation Representation22 References

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Nov 18, 2014Fused Angles for Body Orientation Representation23 Containing set: Parameters:9 ⇒ Redundant Constraints:Orthogonality (determinant +1) Singularities:None Features: Trivially exposes the basis vectors, computationally efficient for many tasks, numerical handling is difficult Rotation Matrices

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Nov 18, 2014Fused Angles for Body Orientation Representation24 Containing set: Parameters:4 ⇒ Redundant Constraints:Unit norm Singularities:None Features: Dual representation of almost every rotation, computationally efficient for many tasks, unit norm constraint must be numerically enforced Quaternions

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